Pick any number with at least three different digits. Jumble the digits however you want to create a different number. Take these two numbers and find their difference. Their difference is your new number. Pick a non-zero digit in your new number and remove it. Give me the rest of the digits in any order you please. From this, I can work out what digit you removed.

How do I find your digit? Why does this work? Prove it.

The idea for this puzzle was taken from a mindreading program here.

You will give me either one or more digits. If you give me two, I'll add them together. If the new number is greater 9, I'll repeat the addition until I get a number less than or equal to nine. If it is nine, the missing digit is nine; otherwise, I will then subtract this number from nine. That is the missing digit.

It works on the same basis as the old "casting out nines" method of checking long sums.  Adding the digits of a number produces another number with the same remainder when divided by 9. Repeating the addition eventually results in a single digit which is that remainder.

Because of this rule, and the commutative property of addition, Two numbers with the same digits, albeit in a different order, leave the same remainder when divided by nine.  Therefore, their difference is divisible by nine -- which means that the digits eventually add up to nine.

It was necessary to specify that the missing digit be non-zero because there is no way to distinguish a missing zero from a missing nine.

Posted by TomM on 2004-03-11 11:15:02